[FOM] Historical Queries on AC

joeshipman@aol.com joeshipman at aol.com
Tue Jan 8 09:51:09 EST 2008

1) In 1938, Tarski (Fund. Math. vol. 30) showed that AC follows from 
the axiom that there is a Universe containing any set (in other words, 
that arbitrarily large inaccessible cardinals exist). Of course, the 
consistency (rather than the truth) of AC doesn't need the full 
Universes axiom, just one inaccessible limit of inaccessibles (because 
that set will satisfy ZF and the Universes axiom).

Was this published before or after Godel's 1938 PNAS paper proving the 
consistency of GCH and AC?

If Tarski's paper came first, there is a sense in which he, not Godel, 
was the first mathematician to provide a consistency proof of AC 
acceptable to modern mathematicians (because the Tarski "Universes 
axiom" is freely used by modern mathematicians in algebraic geometry 
and other core mathematical areas).

2) When was the first proof that GCH implies AC published, and by whom? 
I know that Lindenbaum and Tarski announced the result in 1926, but the 
proof commonly cited comes from a 1947 paper of Sierpinski.

3) Sierpinski's proof is is stronger than just "GCH-->AC", it actually 
shows that for a set A to be well-orderable one needs only that there 
are no intermediate cardinals anywhere in the sequence A < P(A) <  
P(P(A)) < P(P(P(A))) < P(P(P(P(A)))).  Has anyone improved this to 
require a smaller set of no-intermediate-cardinal assumptions?

4) Analogously to 3), what is the best known result on how many levels 
of Universes above A are necessary in order to well-order A?

-- JS
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